The value of the expression C02−C12+C22−……..+(−1)nCn2 is (Here Ck=nCk

When n is odd, take n=2m+1, so thatS=C02−C12+C22−…..+(−1)2mC2mm+(−1)2m+1C2m+12=C02−C2m+12−C12−C2m2+…..+(−1)mCm2−Cm+12 But 2m+1C0+2m+1C2m+1 2m+1C1=2m+1C2m etc. Therefore S=0When n is even, we take n=2 m. In this caseC02−C12+C22−C32+…..+(−1)2mC2m2= Coefficient of constant term inC0−C1x+C2x2−+(−1)2mC2mx2m C0+C11x+C21x2+….+C2m1x2m= Coefficient of constant term in (1−x)2m1+1x2m= Coefficient of x2min⁡(1−x)2m(1+x)2m= Coefficient of x2m in 1−x22m(−1)m 2mCm=(−1)n/2 nCn/2

The value of the expression C02−C12+C22−……..+(−1)nCn2 is (Here Ck=nCk

When n is odd, take n=2m+1, so thatS=C02−C12+C22−…..+(−1)2mC2mm+(−1)2m+1C2m+12=C02−C2m+12−C12−C2m2+…..+(−1)mCm2−Cm+12 But 2m+1C0+2m+1C2m+1 2m+1C1=2m+1C2m etc. Therefore S=0When n is even, we take n=2 m. In this caseC02−C12+C22−C32+…..+(−1)2mC2m2= Coefficient of constant term inC0−C1x+C2x2−+(−1)2mC2mx2m C0+C11x+C21x2+….+C2m1x2m= Coefficient of constant term in (1−x)2m1+1x2m= Coefficient of x2min⁡(1−x)2m(1+x)2m= Coefficient of x2m in 1−x22m(−1)m 2mCm=(−1)n/2 nCn/2

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The value of the expression $C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots \dots . . + (- 1)^{n} C_{n}^{2} is (Here C_{k} =^{n} C_{k})$

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0, if n is odd

$(- 1)^{n}$ , if n is odd

$(- 1)^{n / 2} \cdot C_{n / 2}$ , if n is even

$(- 1)^{n - 1} \cdot C_{n - 1}$ , if n is even

answer is A, C.

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Detailed Solution

When n is odd, take $n = 2 m + 1$ , so that

$\begin{array}{l} S = C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots . . + (- 1)^{2 m} C_{2 m}^{m} + (- 1)^{2 m + 1} C_{2 m + 1}^{2} \\ = (C_{0}^{2} - C_{2 m + 1}^{2}) - (C_{1}^{2} - C_{2 m}^{2}) + \dots . . + (- 1)^{m} (C_{m}^{2} - C_{m + 1}^{2}) \\ {But}^{2 m + 1} C_{0} +^{2 m + 1} C_{2 m + 1} \\ ^{2 m + 1} C_{1} =^{2 m + 1} C_{2 m} etc. Therefore S = 0 \end{array}$

When n is even, we take $n = 2 m$ . In this case

$C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - C_{3}^{2} + \dots . . + (- 1)^{2 m} C_{2 m}^{2}$ = Coefficient of constant term in

$[C_{0} - C_{1} x + C_{2} x^{2} - + (- 1)^{2 m} C_{2 m} x^{2 m}] [C_{0} + C_{1} \frac{1}{x} + C_{2} \frac{1}{x^{2}} + \dots . + C_{2 m} \frac{1}{x^{2 m}}]$